Abstract:
This thesis studies user optimal policies (or user equilibria) for routing in queueing networks, with particular emphasis on networks where the Downs- Thomson paradox appears. We consider here a queueing network with two modes of transport private (road) transportation and public (train) transportation – which are modelled as a single server first-in-first-out ·|M|1 queue and infinite server batch-service ·|G(N)|∞ queue respectively, where N is the size of the batch. The waiting time at the single server queue increases as traffic increases while the service frequency at the batch-service queue increases with demand, and hence waiting time there decreases. User equilibria or user optimal policies may occur when users in a network individually choose the best (shortest) route for themselves. The Downs-Thomson paradox occurs when improvements in service produce an overall decline in performance as user equilibria adjust. We study this network system in the setting of both state-dependent routing and probabilistic routing. We use coupling arguments to show that under state-dependent routing a unique user optimal policy exists and is monotone. We also analyze the user equilibria (user optimal policies) in the setting of probabilistic routing when the road is modelled as an M|G|1 queue instead of an M|M|1 queue. This analysis shows that for some parameter values there exist multiple equilibria (not all stable) for N ≥ 2. Furthermore, we analyze the properties of the user optimal policies for the same network with M|M|1 queue versus M|M(N)|1 queue in the settings of both state-dependent routing and probabilistic routing.